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The main result of this paper is the 4-dimensional supermetric version of theWheeler-DeWitt equation, that uses only one time variable for the both roles -as internal time and for the ADM split, as Hamiltonian evolution parameter. Westudy the ADM split with respect to the scalar massless field serving asinternal time. The 4-dimensional hyper-surfaces $\Sigma_{\phi = const}$ spanthe 5-dimensional space with the scalar field being the fifth coordinate. As aresult we obtain the analog of the Wheeler-DeWitt equation for the4-dimensional supermetric. We compare the ADM action with the non-compactifiedKaluza-Klein action for the same physical space and obtain the equation for theextrinsic curvature and the scalar massless field.

Wheeler-DeWitt Equation for 4D Supermetric and ADM with Massless Scalar Field as Internal Time